PHYSIOLOGICAL MODELING

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PHYSIOLOGICAL MODELING
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OBJECTIVES

• Describe the process used to build a mathematical
  physiological model.
• Explain the concept of a compartment.
• Analyze a physiological system using
  compartmental analysis.
• Solve a nonlinear compartmental model.
• Qualitatively describe a saccadic eye movement.
• Describe the saccadic eye movement system with a
  second-order model.
• Explain the importance of the pulse-step saccadic
  control signal.
• Explain how a muscle operates using a nonlinear
  and linear muscle model.
• Simulate a saccade with a fourth-order saccadic
  eye movement model. Estimate the parameters of a
  model using system identification.

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INTRODUCTION

• Physiology the science of the functioning of
  living organisms and of their component parts.
• 2 Types of physiological model
• A quantitative physiological model is a
  mathematical representation that approximates the
  behavior of an actual physiological system.
• A qualitative physiological model describes the
  actual physiological system without the use of
  mathematics.

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Flow Chart for physiological modeling
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Deterministic and Stochastic Models

• A deterministic model is one that has an exact
  solution that relates the independent variables
  of the model to each other and to the dependent
  variable. For a given set of initial conditions,
  a deterministic model yields the same solution
  each and every time.
• A stochastic model involves random variables that
  are functions of time and include probabilistic
  considerations. For a given set of initial
  conditions, a stochastic model yields a different
  solution each and every time.

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Solutions

• A closed-form solution exists for models that can
  be solved by analytic techniques such as solving
  a differential equation using the classical
  technique or by using Laplace transforms.
• example
• A numerical or simulation solution exists for
  models that have no closed-form solution.
• example

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COMPARTMENTAL MODELING

• Compartmental modeling is analyzing systems of
  the body characterized by a transfer of solute
  from one compartment to another, such as the
  respiratory and circulatory systems.
• It is concerned with maintaining correct chemical
  levels in the body and their correct fluid
  volumes.
• Some readily identifiable compartments are
• Cell volume that is separated from the
  extracellular space by the cell membrane
• Interstitial volume that is separated from the
  plasma volume by the capillary walls that contain
  the fluid that bathes the cells
• Plasma volume contained in the circulatory system
  that consists of the fluid that bathes blood cells

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Transfer of Substances Between Two Compartments
Separated by a Thin Membrane

• Ficks law of diffusion
• Where
• q quantity of solute
• A membrane surface area
• c concentration
• D diffusion coefficient
• dx membrane thickness

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Compartmental Modeling Basics

• Compartmental modeling involves describing a
  system with a finite number of compartments, each
  connected with a flow of solute from one
  compartment to another.
• Compartmental analysis predicts the
  concentrations of solutes under consideration in
  each compartment as a function of time using
  conservation of mass accumulation equals input
  minus output.
• The following assumptions are made when
  describing the transfer of a solute by diffusion
  between any two compartments
• 1. The volume of each compartment remains
  constant.
• 2. Any solute q entering a compartment is
  instantaneously mixed throughout the entire
  compartment.
• 3. The rate of loss of a solute from a
  compartment is proportional to the amount of
  solute in the compartment times the transfer
  rate, K, given by Kq.

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Multi-compartmental Models

• Real models of the body involve many more
  compartments such as cell volume, interstitial
  volume, and plasma volume. Each of these volumes
  can be further compartmentalized.
• For the case of N compartments, there are N
  equations of the general form
• Where qi is the quantity of solute in compartment
  i. For a linear system, the transfer rates are
  constants.

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Modified Compartmental Modeling

• Many systems are not appropriately described by
  the compartmental analysis because the transfer
  rates are not constant.
• Compartmental analysis, now termed modified
  compartmental analysis, can still be applied to
  these systems by incorporating the nonlinearities
  in the model. Because of the non linearity,
  solution of the differential equation is usually
  not possible analytically, but can be easily
  simulated.
• Another method of handling the nonlinearity is to
  linearize the nonlinearity or invoke
  pseudostationary conditions.

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Transfer of Solutes Between Physiological
Compartments by Fluid Flow

• Uses a modified compartmental model to consider
  the transfer of solutes between compartments by
  fluid flow.
• Compartmental model for the transfer of solutes
  between compartments by fluid flow

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Dye Dilution Model

• Dye dilution studies are used to determine
  cardiac output, cardiac function, perfusion of
  organs, and the functional state of the vascular
  system.
• Usually the dye is injected at one site in the
  cardiovascular system and observed at one or more
  sites as a function of time.

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AN OVERVIEW OF THE FAST EYE MOVEMENT SYSTEM

• A fast eye movement is usually referred to as a
  saccade and involves quickly moving the eye from
  one image to another image.
• The saccade system is part of the oculomotor
  system that controls all movements of the eyes
  due to any stimuli.
• Each eye can be moved within the orbit in three
  directions vertically, horizontally, and
  torsionally, due to three pairs of
  agonistantagonist muscles.
• Fast eye movements are used to locate or acquire
  targets.

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TYPES OF EYE MOVEMENTS

• Smooth pursuit - used to track or follow a target
• Vestibular ocular - used to maintain the eyes on
  the target during head movements
• Vergence - used to track near and far targets
• Optokinetic used when moving through a
  target-filled environment or to maintain the eyes
  on target during continuous head rotation
• Visual used for head and body movements

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Saccade Characteristics

• Saccadic eye movements, among the fastest
  voluntary muscle movements the human is capable
  of producing, are characterized by a rapid shift
  of gaze from one point of fixation to another.
• Saccadic eye movements are conjugate and
  ballistic, with a typical duration of 30100ms
  and a latency of 100300ms.

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WESTHEIMER SACCADIC EYE MOVEMENT MODEL

• The first quantitative saccadic horizontal eye
  movement model, was published by Westheimer in
  1954. Westheimer proposed a second-order model
  equation

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THE SACCADE CONTROLLER

• One of the challenges in modeling physiological
  systems is the lack of data or information about
  the input to the system. Recording the signal
  would involve invasive surgery and instrumentation

• In 1964, Robinson attempted to measure the input
  to the eyeballs during a saccade by fixing one
  eye using a suction contact lens, while the other
  eye performed a saccade from target to target. He
  proposed that muscle tension driving the eyeballs
  during a saccade is a pulse plus a step, or
  simply, a pulse-step input .

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THE SACCADE CONTROLLER

• Microelectrode studies have been carried out to
  record the electrical activity in oculomotor
  neurons micropipet used to record the activity
  in the oculomotor nucleus, an important neuron
  population responsible for driving a saccade

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THE SACCADE CONTROLLER

• Collins and his coworkers reported using a
  miniature C-gauge force transducer to measure
  muscle tension in vivo at the muscle tendon
  during unrestrained human eye movements.

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DEVELOPMENT OF AN OCULOMOTOR MUSCLE MODEL

• An accurate model of muscle is essential in the
  development of a model of the horizontal fast eye
  movement system.
• The model elements consist of an active-state
  tension generator (input), elastic elements, and
  viscous elements. Each element is introduced
  separately and the muscle model is incremented in
  each subsection.

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Passive Elasticity

• Involves recording of the tension observed in an
  eye rectus muscle.
• The tension required to stretch a muscle is a
  nonlinear function of distance.

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Active-State Tension Generator

• In general, a muscle produces a force in
  proportion to the amount of stimulation. The
  element responsible for the creation of force is
  the active-state tension generator.
• The relationship between tension, T, active-state
  tension, F, and elasticity is given by

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Elasticity

• Series Elastic Element
• Experiments carried out by Levin and Wyman in
  1927, and Collins in 1975 indicated the need for
  a series elasticity element
• LengthTension Elastic Element
• Given the inequality between Kse and K, another
  elastic element, called the lengthtension
  elastic element, Klt, is placed in parallel with
  the active-state tension element

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ForceVelocity Relationship

• Early experiments indicated that muscle had
  elastic as well as viscous properties.
• Muscle was tested under isotonic (constant force)
  experimental conditions to investigate muscle
  viscosity.

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LINEAR MUSCLE MODEL

• Examines the static and dynamic properties of
  muscle in the development of a linear model of
  oculomotor muscle.
• B- viscosity
• K- elasticity
• F- tension generator

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A LINEAR HOMEOMORPHIC SACCADIC EYE MOVEMENT MODEL

• In 1980, Bahill and coworkers presented a linear
  fourth-order model of the horizontal oculomotor
  plant that provides an excellent match between
  model predictions and horizontal eye movement
  data. This model eliminates the differences seen
  between velocity and acceleration predictions of
  the Westheimer and Robinson models and the data.

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A LINEAR HOMEOMORPHIC SACCADIC EYE MOVEMENT MODEL
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A TRUER LINEAR HOMEOMORPHIC SACCADIC EYE MOVEMENT
MODEL
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SYSTEM IDENTIFICATION

• In modeling physiological systems,
• GOAL not to design a system, but to identify the
  parameters and structure of the system
• Ideally
• Input and output is known
• Information on the Internal dynamics is available

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SYSTEM IDENTIFICATION

• System identification is the process of creating
  a model of a system and estimating the parameters
  of the model.
• 2 concepts of S.I.
• a. Time domain
• b. Frequency domain
• Before S.I. begins, understanding the
  characteristics of the input and output signals
  is important (e.g. voltage and frequency
  range,type of signal whether it is deterministic
  or stochastic and if coding is involved.)

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SYSTEM IDENTIFICATION

• The simplest and most direct method of system
  identification is sinusoidal analysis.
• Source of sinusoidal excitation consists
• a. sine wave generator
• b. a measurement transducer
• c. recorder to gather frequency response
  data(can be obtained using the oscilloscope)

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SYSTEM IDENTIFICATION

• Another type of identification technique either
  for a
• first-order system or
• a second-order system
• is by using a time-domain approach.